The Complete One-loop Spin Chain of N= 1 SQCD

arXiv:1111.5290v1 [hep-th] 22 Nov 2011

Preprint typeset in JHEP style - HYPER VERSION

YITP-SB-11-40

The Complete One-loop Spin Chain of N = 1 SQCD

Pedro Liendoa∗ and Leonardo Rastellia† a

C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794-3840, USA

Abstract: We evaluate the complete planar one-loop dilation operator of N = 1 Super QCD, at the large N Banks-Zaks fixed point near the upper edge of the superconformal window. The spin-chain Hamiltonian turns out to be entirely fixed by the constraints of superconformal symmetry, as in N = 4 Super Yang-Mills and in N = 2 SuperConformal QCD. Keywords: CFT, Integrability.

∗ †

Email: [email protected] Email: [email protected]

Contents 1. Introduction

2

2. Preliminaries 2.1 Superconformal representations

4 5

3. Algebraic Evaluation of the Hamiltonian 3.1 The SU (1, 1) × U (1|1) subsector 3.2 First order expressions for Q(g) and S(g) 3.3 The Hamiltonian as a sum of projectors 3.3.1 V × V and V¯ × V¯ ¯ V¯ × X and X¯ × V 3.3.2 V × X , X¯ × V, 3.3.3 X¯ × X , X˜ × X , X˜ × X¯˜ and X¯ × X¯˜ (gauge contracted) 3.3.4 X × X¯ (flavor contracted), X¯˜ × X˜ (flavor contracted), V × V¯ and V¯ × V. 3.4 Scalar Sector 3.5 Open Chain 3.6 The Harmonic Action ¯ 3.6.1 V × V and V¯ × V. ˜ 3.6.2 V × X , X × V , X¯ × V¯ and V¯ × X¯˜ . 3.6.3 V¯ × X , X˜ × V¯ , X¯ × V and V × X¯˜ . ˜ (gauge contracted). 3.6.4 X¯ × X and X˜ × X¯ ¯ 3.6.5 X˜ × X and X¯ × X˜ . 3.6.6 X × X¯ (flavor contracted), X¯˜ × X˜ (flavor contracted), V × V¯ and V¯ × V.

6 7 8 10 10 11 11 11 12 12 12 13 13 14 14 15 15

4. Spectral analysis 4.1 N = 2 SCQCD 4.1.1 Parity 4.1.2 Diagonalization 4.2 N = 1 SQCD

16 16 17 18 20

5. Discussion

21

A. N = 1 superconformal multiplets

23

B. Oscillator Representation B.1 Vector multiplets V and V¯ B.2 Chiral multiplets X and X¯ B.3 Two-letter Superconformal Primaries

23 24 24 25

–1–

1. Introduction Superconformal field theories have come to occupy a central position in theoretical physics. Much has been learnt, especially about the most supersymmetric models in four and three dimensions, but some of the most interesting questions remain unanswered. To mention a glaring example, which is the focus of this paper, no substantial progress has been made over the years on the simplest four-dimensional N = 1 model originally discovered by Seiberg [1]: SU (Nc ) super QCD in the conformal window. No weakly-coupled AdS gravity dual description is expected to exist for this class of theories; non-critical string duals (which have an intrinsically strongly-coupled worldsheet description) may well exist (see e.g. [2–6]), but these proposals are not yet well understood. And Seiberg duality remains largely a mystery.1 A strategy that has proved very fruitful in the case of N = 4 super Yang-Mills theory is the systematic perturbative evaluation of the planar dilation operator [10–16], which turns out to be described by an integrable spin chain (see e.g. [17–22] for a very partial list of references and [23] for a comprehensive review). One does not a priori expect the remarkable integrability properties of N = 4 SYM to fully persist in less symmetric theories, but one cannot really know until one tries.2 For N = 1 SQCD, a perturbative expansion is meaningful in the large N Veneziano limit [28] of Nc → ∞, Nf → ∞ with Nf /Nc fixed, near the upper edge of the superconformal window Nf . 3Nc . If one defines Nf = 3− ǫ, Nc

(1.1)

the large N theory flows for ǫ ≪ 1 to a weakly-coupled Banks-Zaks fixed point [29], with ’t Hooft coupling gY2 M Nc ∼ ǫ. The one-loop planar dilation operator captures the spectrum of the theory at this isolated fixed point, while higher-loop corrections (reorganized in powers of ǫ) correspond to moving down the conformal window. The dual “magnetic” theory admits a perturbative expansion starting from the lower edge of the conformal window Nf & 32 Nf , with a Banks-Zaks fixed point that is weakly-coupled for ǫ˜ ≪ 1, where Nf 3 = + ǫ˜ . Nc 2

(1.2)

A complete large N solution of SQCD would entail determining the dilation operator of the electric theory to all orders in ǫ, and that of the magnetic theory to all orders in ˜ǫ. The resummations of both expansions should then coincide – in the ultimate triumph of Seiberg 1 We should however mention the new precision checks of Seiberg duality for the protected spectrum, obtained with the help of the superconformal index [7–9]. 2 There has been much work on integrability in 4d gauge theories with N < 4, see e.g. [24–26] and the recent review [27]. Most investigations to date have focused on theories in the standard ’t Hooft limit, where the number of fundamental flavors Nf is zero or anyway kept fixed as the number of colors Nc is sent to infinity. The ’t Hooft limit has the drawback that conformal invariance is inevitably broken (except of course in N = 4 SYM). The breaking of conformal invariance does not affect the one-loop dilation operator, but it kicks in at two loops.

–2–

duality. Needless to say, this is a tall order, and one cannot hope to fulfill this program unless integrability comes to the rescue. Recently the one-loop dilation operator of N = 1 SQCD in the Veneziano limit has been determined in the scalar subsector [30], and shown to coincide with the Ising spin chain in a transverse magnetic field, one of the best known integrable models. This is a tantalizing hint, well-worth subjecting to more stringent tests. In this paper we determine the complete planar one-loop spin chain Hamiltonian of N = 1 SQCD and make a preliminary investigation of its integrability. We eschew direct Feynman diagram calculations and rely instead on symmetry. The constraints of superconformal invariance are sufficient to completely fix the one-loop planar dilation operator of N = 4 super Yang-Mills [12, 31], and also, somewhat unexpectedly, of N = 2 superconformal QCD, as recently shown in [32]. The calculation for N = 1 SQCD proceeds along similar lines as N = 2 SCQCD, and again we are able to fix the one-loop Hamiltonian from symmetry considerations alone. This is a nice surprise. For N = 4 SYM at each site of the chain sits a single irreducible multiplet, which moreover has a simple tensor product with itself; by contrast in the N = 2 and N = 1 cases each site hosts a handful of irreducible representations, and their tensor products have a more complicated decomposition, leading to a rather intricate mixing problem. Despite these complications, the general structure of the calculation is the same as in N = 4 SYM: the full Hamiltonian can be uplifted from the Hamiltonian in a simple subsector, which is in turn uniquely fixed by a centrally extended SU (1|1) symmetry. The rest of the paper is organized as follows. We begin in section 2 by introducing N = 1 SQCD and its spin chain. We discuss the decomposition of the single-letter state space into irreducible representations of the N = 1 superconformal algebra and give the tensor products of any two such representations. In section 3 we show how the full planar one-loop Hamiltonian can be uplifted from a closed subsector with SU (1, 1) × U (1|1) symmetry, where it is fixed (up to overall normalization) by the centrally extended superalgebra. By this route we obtain an expression for the full Hamiltonian in terms of the superconformal projectors onto the different irreps that appear in the two-site state space. By making contact with the scalar sector results of [30] we fix the overall normalization. Finally we obtain the “harmonic action” form of the full Hamiltonian – a completely explicit oscillator representation that can be easily implemented on any state. In section 4 we present a preliminary investigation of the integrability of the one-loop spin chains for N = 2 SCQCD and N = 1 SQCD. We show that both models admit a “parity” symmetry that commutes with the Hamiltonian to all loops. In N = 4 SYM, the existence of degenerate parity pairs provided early circumstantial evidence for integrability. We diagonalize the spectrum and search for parity pairs, both in the N = 2 and N = 1 spin chains (in the subsectors that were used to uplift the full Hamiltonians, for states of length L ≤ 5). We find that the appearance of parity pairs is much less systematic than in N = 4 SYM. We conclude in section 5 with a brief discussion. Two appendices contain background material on the N = 1 superconformal algebra and its oscillator representation, and the explicit expressions of the two-site superconformal primaries.

–3–

2. Preliminaries We consider N = 1 SQCD, the N = 1 supersymmetric Yang-Mills theory with gauge group SU (Nc ) and Nf flavors of fundamental quarks. In Table 1 we recall the familiar symmetries of the theory and set our notations. Besides the N = 1 vector multiplet (Aαα˙ , λα ), in the adjoint representation of the gauge group, there are two sets of Nf chiral multiplets, (Q, ψα ) ˜ ψ˜α ), respectively in the fundamental and antifundamental representations of SU (Nc ). and (Q, The color and flavor structure is then (Aab , λab ) ,

(Qai , ψ ai ) ,

˜ a˜ı , ψ˜a˜ı ) , (Q

(2.1)

where a = 1, . . . Nc are color indices, and i = 1, . . . Nf and ˜ı = 1, . . . Nf two independent sets of flavor indices, corresponding to the independent flavor symmetries of the gauge-fundamental and of the gauge-antifundamental chiral multiplets. SU (Nc )

SU (Nf )

SU (Nf )

U (1)B

U (1)r

Qα

1

1

1

0

Sα

1

1

1

0

−1

λα

Adj

1

1

0

1

Aαα˙

Adj

1

1

0

0

Q





1

1

ψα ˜ Q





1

1

1−



1



ψ˜α

−1



1



−1

1

Nc Nf Nc −N f Nc 1− N f Nc − Nf

˙ for Lorentz spinor Table 1: Field content and symmetries of N = 1 SQCD. We use α = ± and α˙ = ± α indices. Qα and S denote respectively the Poincaré and conformal supercharges. Conjugate objects ¯α˙ are not written explicitly. such as λ

In the large N Veneziano limit, the basic flavor-singlet local gauge-invariant operators are “generalized single-traces” [33, 34], of the schematic form   Tr φk1 Mk2 φk3 Mk4 . . . . (2.2) Here φ denotes any of the color-adjoint “letters”, for example φab = (D n λ)ab , where D is a gaugecovariant derivative, while Mab is any of the gauge-adjoint composite objects obtained by the ¯ bi flavor contraction of a fundamental and an antifundamental letter, for example Mab = Qai Q ¯ ˜ or Mab = ψ˜a˜ı Q b˜ı . In the Veneziano limit these are the building blocks: a generic gauge-singlet operator factorizes into products of generalized single-trace operators, up to 1/N corrections. To leading order in the large N limit (the planar theory) generalized single-trace operators are closed under the action of the dilation operator, which takes the familiar form of a spin chain

–4–

Hamiltonian; as usual, the locality of the Hamiltonian (nearest neighbor, next-to-nearestneighbor, ...) is related to the loop order of the planar perturbative expansion. One can also consider operators with open flavor indices of the schematic forms q¯i φk1 Mk2 . . . q j ,

q¯i φk1 Mk2 . . . q¯˜˜ ,

q˜˜ı φk1 Mk2 . . . q j ,

q˜˜ı φk1 Mk2 . . . q¯˜˜ ,

(2.3)

where q and q˜ stand for any of the (anti)fundamental letters. It is also understood that for operators with index structures i j and ˜ı ˜ we are projecting into the adjoint representation of SU (Nf ) by removing the flavor-trace term. In the Veneziano limit operators of the form (2.3) are closed under renormalization and they can be viewed as open spin chains. 2.1 Superconformal representations We are going to make crucial use of superconformal symmetry to constraint the form of the one-loop dilation operator (the spin chain Hamiltonian). The letters that occupy each site of the N = 1 SQCD chain belong to four distinct irreducible representations of the SU (2, 2|1) superconformal algebra. We denote them by X (chiral multiplet), X¯ (antichiral multiplet), V (vector multiplet) and V¯ (conjugate vector multiplet). This is in contrast with N = 4 SYM, where the letters belong to a single irreducible representation, but rather similar to N = 2 SCQCD, where the single-site state space decomposes in the sum of three different representations [32]. At one loop, the Hamiltonian is of nearest-neighbor form. The Hamiltonian density acts on two adjacent sites and can be written as a sum of projectors onto the irreducible representations that span the two-site state space. Because of the index structure of the spin chain, not all orderings of two single-site representations are allowed in the two-site state space. For example, it is not possible to have two Qs adjacent to each other because there is no way in which to contract the indices, so two adjacent X representations are not allowed. ¯ can be placed together and, in fact, there are two ways in which On the other hand, Q and Q this can be done, we can contract either adjacent gauge indices or adjacent flavor indices. The gauge-contracted combinations are (the order matters): V ×V

V ×X ˜ V × X¯

X¯ × X

V¯ × V X˜ × V

V¯ × V¯ V¯ × X

V¯ × X¯˜

X¯ × V ¯ X¯ × X˜

X˜ × X

V × V¯ X˜ × V¯

X¯ × V¯ X˜ × X˜¯

(2.4) (2.5) (2.6) ,

(2.7)

while the flavor-contracted combinations are: X¯˜ × X˜ .

X × X¯

(2.8)

For clarity we have added a “tilde” to distinguish the fundamental from the antifundamental chiral multiplets, though of course this is a distinction that pertains to the color and fla-

–5–

vor structure, not the superconformal structure (X and X˜ are isomorphic as superconformal representations). The classification of multiplets of the N = 1 superconformal algebra is reviewed in Appendix A. We follow the notations of [9], according to which the multiplets that span the single-site state space are given by ¯ (0,0) , X = X˜ = D

X¯ = X¯˜ = D(0,0) ,

¯ 1 , V =D ( ,0)

V¯ = D(0, 1 ) .

2

2

(2.9)

Using superconformal characters it is not difficult to decompose the tensor products of any two such multiplets into irreducible representations. We find X˜ × X = B¯ 4 (0,0) ⊕ 3

∞ M

q=0 ∞ M

˜=B 4 X¯ × X¯ − (0,0) ⊕ 3

X × X¯ = X¯ × X =

∞ M q=0

X¯ × V =

q=0 ∞ M q=0

∞ M q=1

Cˆ( q , q+1 ) , 2

Cˆ( q+1 , q−1 ) = X˜ × V , 2

2

(2.13)

Cˆ( q+1 , q ) = V × X¯˜ ,

(2.15)

∞ M

(2.16)

2

2

2

2

q=1

Cˆ( q−1 , q+1 ) = V¯ × X¯˜ ,

V × V = B¯2(0,0) ⊕ B¯2(1,0) ⊕

2

2

∞ M

Cˆ( q+1 , q−2 ) ,

q=2

V¯ × V¯ = B−2(0,0) ⊕ B−2(0,1) ⊕

q=1

(2.12)

(2.14)

2

3

∞ M

(2.11)

2

Cˆ( q , q+1 ) = X˜ × V¯ ,

X¯ × V¯ = B− 5 (0, 1 ) ⊕

V × V¯ =

(2.10)

2

2 2

3 2

∞ M

q=0

2

Cˆ( q , q ) = X˜ × X¯˜ = X¯˜ × X˜ ,

V × X = B¯ 5 ( 1 ,0) ⊕ V¯ × X =

Cˆ( q+1 , q ) ,

Cˆ( q , q ) = V¯ × V . 2 2

∞ M q=2

2

2

Cˆ( q−2 , q+1 ) , 2

2

(2.17) (2.18) (2.19)

3. Algebraic Evaluation of the Hamiltonian The evaluation of the one-loop Hamiltonian proceeds in three steps. First, we identify a closed SU (1, 1) × U (1|1) subsector and determine the Hamiltonian in the subsector by using the constraints of the centrally-extended symmetry. We then uplift the result to the complete

–6–

theory and obtain the full Hamiltonian as a sum of superconformal projectors. Finally we rewrite the Hamiltonian in an explicit “harmonic action” form. 3.1 The SU (1, 1) × U (1|1) subsector Consider the subsector generated by the letters Dk λ+ , k! Dk Qk = Q, k! k ˜, ˜k = D Q Q k!

λk =

Dk ¯ F¯k = F˙ ˙ , k! ++ Dk ¯ ψ¯k = ψ˙ , k! + D k ¯˜ ψ˙ . ψ¯˜k = k! +

(3.1) (3.2) (3.3)

with D ≡ D++˙ . By using conservation of the engineering dimension, of the Lorentz spins and of the R-charge it is easy to see that this sector is closed to all loops under the action of the dilation operator. Moreoever, the one-loop Hamiltonian restricted to this subsector can be uplifted to the full one-loop Hamiltonian, as each of the modules appearing on the right hand side of the tensor products (2.10–2.19) contains a representative within the subsector. The representatives are primaries of SU (1, 1), and descendants with respect to the full SU (2, 2|1) algebra. To obtain the Hamiltonian in the subsector one could perform an explicit Feynman diagram calculation. Instead we use a purely algebraic approach that uses the restrictions imposed by superconformal symmetry. The algebraic method was successfully used in [31] and [32] to find the dilation operator of N = 4 SYM and N = 2 SCQCD respectively. The subgroup of the superconformal group acting on the sector is SU (1, 1) × U (1|1). The SU (1, 1) generators are J+′ (g) = P++˙ (g) ,

(3.4)

˙

J−′ (g) = K++ (g) , 1 1 1 1 ˙ J3′ (g) = D0 + δD(g) + L++ + L˙ +˙ + . 2 2 2 2

(3.5) (3.6)

The U (1|1) generators are given by ˙

L = L−− + L˙ −˙ − + D0 ,

¯ ¯ ˙ (g) , Q(g) =Q −

¯ S(g) = S¯−˙ (g) ,

δD(g)

(3.7)

and can be checked to commute with the SU (1, 1) generators (3.4–3.6). Their (anti)commutators are3 ¯ ¯ , [L, Q(g)] = Q(g)

(3.8)

¯ and S¯ does not denote complex conjugation, we are going to impose the appropriate The bar in Q hermiticity condition below. 3

–7–

¯ ¯ , [L, S(g)] = −S(g) 1 ¯ ¯ {S(g), Q(g)} = δD(g) . 2

(3.9) (3.10)

¯ increases the The generator L can be identified with the length operator, this implies that Q length of a chain while S¯ decreases it. 3.2 First order expressions for Q(g) and S(g) The procedure is now very similar to the one followed in [32] for N = 2 SCQCD and we shall be brief. Writing Q(g) = gQ + O(g2 ), we formulate an ansatz for the action of the supercharges on the states of the sector compatible with the quantum numbers of the fields ¯ = 0 and [J ′ , S] ¯ = 0. In fact strict and impose invariance under the SU (1, 1) algebra, [J ′ , Q] invariance is too restrictive and one needs only to impose vanishing of these commutators up to local gauge transformations on the chain. It can be easily checked that the following transformations evaluate to zero on any closed chain, ¯ n = α (λn λ + λλn ) , [J+′ , Q]λ
¯ F¯n = α −F¯n λ + λF¯n , [J+′ , Q] ¯ n = αλQn , [J ′ , Q]Q

(3.11) (3.12) (3.13)

+ ¯ ψ¯n [J+′ , Q] ¯Q ˜n [J+′ , Q]

= αψ¯n λ , ˜ nλ , = −αQ ¯ ψ¯˜n = αλψ¯˜n , [J+′ , Q]

(3.14) (3.15) (3.16)

¯ consistent with these transformawhere α is an arbitrary gauge parameter. The action of Q tions is ¯ n =α Qλ ¯ F¯n = α Q

n−1 X

(k′

k ′ =0 n−1  X

1 1 ¯ Fk′ λn−k′ −1 + ′ λk′ F¯n−k′ +1 − n − k′ k +1

k ′ =0 n X ′

+α ¯ n =α QQ ¯ ψ¯n = α Q

k ′ =0 n−1 X

k ′ =0 n−1 X

Qik′ ψ¯n−k′ i

n X ˜ n−k′ ˜ı , +α ψ¯˜˜kı ′ Q ′′

(3.17)  (3.18)

k ′ =0

1 λk′ Qn−k′ −1 , +1

(3.19)

1 ¯ ψk′ λn−k′ −1 , n − k′

(3.20)

k′

k ′ =0 n−1 X

¯Q ˜ n = −α Q

n+1 λk′ λn−k′ −1 , + 1)(n − k′ )

k ′ =0

1 ˜ Qk′ λn−k′ +1 , n − k′

–8–

(3.21)

˜n = α ¯ ψ¯ Q

n−1 X

k ′ =0

k′

1 λk′ ψ¯n−k′ +1 . +1

(3.22)

The terms with α′ and α′′ are invariant on their own and that’s why we assigned them independent gauge parameters. Similarly, the gauge transformations associated with the S¯ supercharge are ¯ k λn−k = β (δk=0 + δn=k ) λn , [J−′ , S]λ ¯ k F¯n−k = βδk=0 F¯n , [J−′ , S]λ ¯ k Qn−k = βδk=0 Qn , [J ′ , S]λ

(3.23) ¯ F¯k λn−k = −βδn=k F¯n , [J−′ , S] ¯Q ˜ k λn−k = −βδn=k Q ˜n , [J−′ , S] ¯ k ψ¯˜n−k = βδk=0 ψ¯˜n , [J−′ , S]λ

−

¯ ψ¯k λn−k = βδn=k ψ¯n , [J−′ , S]

(3.24) (3.25) (3.26)

and the action of S¯ consistent with them is ¯ k λn−k = βλn+1 , Sλ

(3.27)

(n − k + 2)(n − k + 1) ¯ Sλk F¯n−k = β Fn+1 , (n + 3)(n + 2) (k + 2)(k + 1) ¯ Fn+1 , S F¯k λn−k = −β (n + 3)(n + 2) ¯ k Qn−k = βQn+1 , Sλ

˜ k λn−k = −β Q ˜ n+1 , S¯Q k+1 ¯ ψn+1 , S¯ψ¯k λn−k = β n+2 ˜n−k = β n − k + 1 ψ¯˜n+1 , ¯ k ψ¯ Sλ n+2 n−k+1 ¯ i ¯ ′ Fn , SQk ψn−k i = β (n + 2)(n + 1) k+1 ′′ ˜˜ı Q ˜ S ψ¯ F¯n . k n−k ˜ı = β (n + 2)(n + 1)

(3.28) (3.29) (3.30) (3.31) (3.32) (3.33) (3.34) (3.35)

As before, the terms with β ′ and β ′′ are invariant on their own. Note that the action of S¯ ¯ ˜ on Qk ψ¯n−k and ψ˜k Q n−k is non-zero only for the flavor-contracted combinations. Indeed the action on the gauge-contracted combinations would have to give a single letter with open flavor indices which is impossible. Now we impose the hermiticity condition ¯ † = S¯ , Q

(3.36)

which implies the following reality constraints for the undetermined coefficients: α = β∗ ,

α′ = β ′∗ ,

–9–

α′′ = β ′′∗ .

(3.37)

3.3 The Hamiltonian as a sum of projectors ¯ ¯ Having determined the O(g) action of Q(g) and S(g), the one-loop Hamiltonian is easily obtained from ¯ Q} ¯ . H ′ = 2{S, (3.38) This result suffers from a certain gauge ambiguity, as we may redefine ′ ′ − Kℓ + Kℓ+1 , → Hℓ,ℓ+1 Hℓ,ℓ+1

(3.39)

′ is the Hamiltonian density at sites (ℓ, ℓ + 1) and Kℓ an arbitrary local operator where Hℓ,ℓ+1 P ′ at site ℓ. The total Hamiltonian H ′ = ℓ Hℓ,ℓ+1 is guaranteed to be invariant under SU (1, 1). As in [32], we fix the gauge ambiguity (3.39) by demanding the stronger condition that the ′ as a sum of Hamiltonian density be SU (1, 1) invariant as well. We can then write Hℓ,ℓ+1 projectors onto the SU (1, 1) irreps of the two-site state space, with coefficients determined by explicit evaluation on the primary of each module. The uplifting procedure is straightforward: one writes the full Hamiltonian as a sum over SU (2, 2|2) projectors and fixes the coefficients by comparison with the SU (1, 1) subsector, as each SU (1, 1) primary is also a SU (2, 2|2) descendant. We simply quote the results in the various subspaces.

3.3.1 V × V and V¯ × V¯ We find H12 = 0 × PB¯2(0,0) + 2|α|2

∞ X

2h(q)P( q+1 , q−2 ) ,

q=1

2

2

(3.40)

for V × V and ∞

H12 = 0 × PB−2(0,0) + 2(−2|α|2 +

|α′ |2 |α′′ |2 X 2h(q)P( q−2 , q+1 ) , + ) 2 2 2 2

(3.41)

q=1

¯ CPT invariance implies that these expressions should be identical, which imposes for V¯ × V. an extra restriction on |α′ |2 and |α′′ |2 namely, |α′ |2 + |α′′ |2 = 6|α|2 .

(3.42)

Now, α′ and α′′ are parameters associated with (X , X¯ ) and (X˜ , X¯˜ ) repectively. Parity (which is a symmetry of the theory, see section 4) interchanges the two, in order to have parity invariant Hamiltonian we need to set √

3eiθ α , √ α′′ = − 3eiθ α , α′ =

where θ is an arbitrary phase, which can be set to zero by a similarity transformation.

– 10 –

(3.43) (3.44)

¯ V¯ × X and X¯ × V 3.3.2 V × X , X¯ × V, The Hamiltonian in these subspaces is4 V × X = 2|α|2

X

1 (h(q + 1) + h(q) − )P( q+1 , q−1 ) = X˜ × V , 2 2 2

(3.45)

X¯ × V¯ = 2|α|2

X

1 (h(q + 1) + h(q) − )P( q−1 , q+1 ) = V¯ × X¯˜ , 2 2 2

(3.46)

q=0

q=0

1 (h(q + 2) + h(q) − )P( q , q+1 ) = X˜ × V¯ , 2 2 2 q=0 X 1 (h(q + 2) + h(q) − )P( q+1 , q ) = V × X¯˜ . X¯ × V = 2|α|2 2 2 2 V¯ × X = 2|α|2

X

(3.47) (3.48)

q=0

˜ and X¯ × X¯˜ (gauge contracted) 3.3.3 X¯ × X , X˜ × X , X˜ × X¯ Here we find X¯ × X = 2|α|2

X (h(q + 1) + h(q) − 1)P( q , q ) = X˜ × X¯˜ ,

(3.49)

X˜ × X = 2|α|2

X

(2h(q + 1) − 1)P( q+1 , q ) ,

(3.50)

X ˜ = 2|α|2 (2h(q + 1) − 1)P( q , q+1 ) . X¯ × X¯

(3.51)

2 2

q=0

2

q=−1

2

q=−1

2

2

3.3.4 X × X¯ (flavor contracted), X˜¯ × X˜ (flavor contracted), V × V¯ and V¯ × V. This is the most involved subspace since we have mixing between different copies of the same superconformal multiplets. The Hamiltonian acting on this subspace is a 4 × 4 matrix,  H12 =

 X¯X 

¯ X˜ V V¯ VV ¯ X X¯ X˜ −1

0

1

0

0

0

0

0

0

0

1

 3|α|2  X˜X¯˜ −1  ¯  VV

V V¯



 ∞  X¯ X X  ˜ ¯˜ 2 + 2|α|  XX  q=1  VV ¯

√

0



   0  P(0,0)  0 

(3.52)

0

X X¯

˜ X˜ X¯

0

0

0 iθ

e 3 q(q+2)

√ eiθ V V¯ − 3 q(q+1)(q+2)

√

V V¯

3e−iθ √ e−iθ − 3 q+1

¯ VV √ e−iθ − 3 q+1 √ −iθ 3e

0 √ 2 eiθ − 3 q(q+1)(q+2) h(q + 2) + h(q − 1) q(q+1)(q+2) √ eiθ 2 3 q(q+2) h(q + 2) + h(q − 1) q(q+1)(q+2)

4



   q q  P( 2 , 2 ) .  

Some of the modules with low q are not present in the subsector so the corresponding coefficients are not determined by the algebraic constraints. However, these coefficients can be fixed by invoking CPT.

– 11 –

3.4 Scalar Sector Let us compare our results with the scalar sector computation of [30]. Apart from providing a check of our procedure, this comparison allows us to fix the overall normalization of the Hamiltonian in terms of the gauge theory ’t Hooft coupling. The action of the Hamiltonian ˜ pairs can be obtained from (3.49) at q = 0 and (3.50,3.51) for the gauge contracted Q and Q at q = −1,   ¯˜ QQ ¯˜ ¯ Q ¯Q ˜ Q ˜Q QQ    QQ  ¯ 0     2 , ˜ H12 = 2|α|  QQ (3.53) −1   ¯  Q  ˜¯ −1  Q  ¯˜ Q ˜ Q 0 in perfect agreement with equation (3.6) of [30] provided we identify 2|α|2 = λ. For the flavor contracted pairs we obtain from the first matrix in (3.52),

H12

 ¯ Q ˜¯ Q ˜ QQ    ¯ = 3|α|2   QQ 1 −1  , ¯˜ −1 1 ˜Q Q 

(3.54)

again in agreement with [30]. It is interesting that the value of the transverse magnetic field for the Ising spin chain in the scalar sector, namely hIsing = Nf /Nc = 3 [30], turns out to be determined by superconformal symmetry alone. 3.5 Open Chain The extension of the previous results to an open chain with adjoint or bi-fundamental flavor indices is straightforward. To obtain the full Hamiltonian we start considering the open chain states of the SU (1, 1) subsector, ψ¯ai . . . Qaj ,

¯ ψ¯ai . . . ψ˜a˜ ,

˜ a˜ı . . . Qaj , Q

˜ a˜ı . . . ψ¯˜a˜ , Q

(3.55)

where i j , ˜ı ˜ are projected to its adjoint irreducible component (the singlet is subtracted). One easily check that the gauge transformations (3.11–3.16) and (3.27–3.35) leave invariant open chains of this form. The uplift to the full Hamiltonian works just as for the closed chain. The upshot is that the Hamiltonian density derived for the closed chain applies with no modification to the open chains. 3.6 The Harmonic Action In this section we present an explicit oscillator form of the Hamiltonian analogous to Beisert’s harmonic action [12] for N = 4 SYM.

– 12 –

¯ 3.6.1 V × V and V¯ × V. For states in these two subspaces the action of the Hamiltonian is identical with that of N = 4 SYM and of N = 2 SCQCD. General states in V × V and V¯ × V¯ can be written as |s1 , ..., sn ; AiV × V = A†s1 A1 ...A†sn An |0i ⊗ |0i ,

|s1 , ..., sn ; AiV¯ × V¯ = A†s1 A1 ...A†sn An |d3 i ⊗ |d3 i ,

(3.56) (3.57)

where A†A = (a†α , b†α˙ , c† ) and si = 1, 2 indicates in which site the oscillator sits. The action of the Hamiltonian on this state does not change the number of oscillators but merely shifts them from site 1 to site 2 (or vice versa) in all possible combinations. This can be written as H12 |s1 , ..., sn ; AiV × V = H12 |s1 , ..., sn ; AiV¯ × V¯ =

X

cn,n12 ,n21 δC1 ,0 δC2 ,0 |s′1 , ..., s′n ; AiV × V ,

(3.58)

X

cn,n12 ,n21 δC1 ,0 δC2 ,0 |s′1 , ..., s′n ; AiV¯ × V¯ ,

(3.59)

s′1 ,...,s′n

s′1 ,...,s′n

where the Kronecker deltas project onto states with zero central charge and nij counts the number of oscillators moving from site i to site j. The explicit formula for the function cn,n12 ,n21 is cn,n12 ,n21 = (−1)1+n12 n21

Γ( 21 (n12 + n21 ))Γ(1 + 12 (n − n12 − n21 )) , Γ(1 + 12 n)

(3.60)

with cn,0,0 = h( n2 ). In [12] it was proven that this function is a superconformal invariant and that it has the appropriate eigenvalues when acting on the Cˆ( q+1 , q−2 ) and Cˆ( q−2 , q+1 ) modules, 2 2 2 2 namely H12 Cˆ( q+1 , q−2 ) = 2h(q)Cˆ( q+1 , q−2 ) , 2

2

2

2

H12 Cˆ( q−2 , q+1 ) = 2h(q)Cˆ( q−2 , q+1 ) . 2

2

2

2

(3.61) (3.62)

3.6.2 V × X , X˜ × V , X¯ × V¯ and V¯ × X¯˜ . General states in these four subspaces can be written as |s1 , ..., sn ; AiV × X = A†s1 A1 ...A†sn An |0i ⊗ |d1 i ,

˜ 1 i ⊗ |0i , |s1 , ..., sn ; AiX˜ × V = A†s1 A1 ...A†sn An |d

|s1 , ..., sn ; AiX¯ × V¯ = A†s1 A1 ...A†sn An |d1 d2 i ⊗ |d3 i ,

˜ 1d ˜ 2i , |s1 , ..., sn ; AiV¯ × X¯˜ = A†s1 A1 ...A†sn An |d3 i ⊗ |d

– 13 –

(3.63) (3.64) (3.65) (3.66)

where |di i = d†i |0i 5 . The action of H12 for all these four subspaces is given by H12 |s1 , ..., sn ; Ai =

X

s′1 ,...,s′n

6

1 cn+1,n12 ,n21 |s1 , ..., sn ; Ai − |s1 , ..., sn ; Ai . 2

(3.67)

Invariance under the superconformal group is guaranteed because we are using the cn,n12 ,n21 constants. We need only to check that this expression has the correct eigenvalues when acting on the corresponding N = 1 primaries (see Appendix B.3), which can be easily done using an algebra software like Mathematica.

3.6.3 V¯ × X , X˜ × V¯ , X¯ × V and V × X¯˜ . In this case we have |s1 , ..., sn ; AiV¯ × X = A†s1 A1 ...A†sn An |d3 i ⊗ |d1 i ,

(3.68)

|s1 , ..., sn ; AiX˜ × V¯ =

(3.69)

|s1 , ..., sn ; AiX¯ × V = |s1 , ..., sn ; AiV × X¯˜ =

˜ 1 i ⊗ |d3 i , A†s1 A1 ...A†sn An |d A†s1 A1 ...A†sn An |d1 d2 i ⊗ |0i , ˜ 1d ˜ 2i , A†s1 A1 ...A†sn An |0i ⊗ |d

(3.70) (3.71)

and the action of H12 reads H12 |s1 , ..., sn ; Ai =

X

s′1 ,...,s′n

1 cn+2,n12 ,n21 |s1 , ..., sn ; Ai − |s1 , ..., sn ; Ai . 2

(3.72)

˜ (gauge contracted). 3.6.4 X¯ × X and X˜ × X¯ The states are given by |s1 , ..., sn ; AiX¯ × X = A†s1 A1 ...A†sn An |d1 d2 i ⊗ |d1 i ,

(3.73)

|s1 , ..., sn ; AiX˜ × X¯˜ =

(3.74)

˜ 1i A†s1 A1 ...A†sn An |d

˜ 1d ˜ 2i , ⊗ |d

and the action of H12 is H12 |s1 , ..., sn ; Ai =

X

s′1 ,...,s′n

cn+1,n12 ,n21 |s1 , ..., sn ; Ai − |s1 , ..., sn ; Ai .

(3.75)

The tilde in some of the d oscillators is just a reminder that we are looking at the X˜ multiplet or its conjugate. 6 To simplify the notation we will omit the Kronecker deltas δC1 ,0 δC2 ,0 . 5

– 14 –

˜. 3.6.5 X˜ × X and X¯ × X¯ The states are given by ˜ 1 i ⊗ |d1 i , |s1 , ..., sn ; AiX˜ × X = A†s1 A1 ...A†sn An |d

(3.76)

|s1 , ..., sn ; AiX¯ × X¯˜ =

(3.77)

A†s1 A1 ...A†sn An |d1 d2 i

˜ 1d ˜ 2i , ⊗ |d

and the action of H12 is H12 |s1 , ..., sn ; Ai =

X

s′1 ,...,s′n

(cn+2,n12 ,n21 − cn+2,n12 +1,n21 +1 ) |s1 , ..., sn ; Ai − |s1 , ..., sn ; Ai . (3.78)

3.6.6 X × X¯ (flavor contracted), X˜¯ × X˜ (flavor contracted), V × V¯ and V¯ × V. The states are |s1 , ..., sn ; AiX × X¯ = A†s1 A1 ...A†sn An |d1 i ⊗ |d1 d2 i ,

(3.79)

|s1 , ..., sn ; AiX¯˜ × X˜ =

(3.80)

|s1 , ..., sn ; AiV × V¯ = |s1 , ..., sn ; AiV¯ × V =

˜ 1d ˜ 2 i ⊗ |d ˜ 1i , A†s1 A1 ...A†sn An |d A†s1 A1 ...A†sn An |0i ⊗ |d3 i , A†s1 A1 ...A†sn An |d3 i ⊗ |0i .

(3.81) (3.82)

The action of H12 is given by H12 |s1 , ..., sn ; AiX × X¯ =

X

s′1 ,...,s′n

cn+3,n12 +1,n21 +1 |s1 , ..., sn ; AiX × X¯

X

+3

s′1 ,...,s′n

cn+3,n12 +1,n21 +2 |s1 , ..., sn ; AiX¯˜ × X˜

X √ cn+3,n12 ,n21 +1 |s1 , ..., sn ; AiV × V¯ + 3e−iθ s′1 ,...,s′n

√

X

+ 3e−iθ

s′1 ,...,s′n

H12 |s1 , ..., sn ; AiX¯˜ × X˜ =

X

s′1 ,...,s′n

−3

cn+3,n12 +1,n21 +1 |s1 , ..., sn ; AiV¯ × V ,

(3.83)

cn+3,n12 +1,n21 +1 |s1 , ..., sn ; AiX¯˜ × X˜

X

s′1 ,...,s′n

cn+3,n12 +1,n21 +2 |s1 , ..., sn ; AiX × X¯

X √ + 3e−iθ cn+3,n12 +1,n21 +1 |s1 , ..., sn ; AiV × V¯ √

s′1 ,...,s′n

+ 3e−iθ

X

s′1 ,...,s′n

– 15 –

cn+3,n12 +1,n21 |s1 , ..., sn ; AiV¯ × V ,

(3.84)

H12 |s1 , ..., sn ; AiV × V¯ =

X

s′1 ,...,s′n

+

cn+3,n12 ,n21 |s1 , ..., sn ; AiV × V¯

X

s′1 ,...,s′n

cn+3,n12 +2,n21 +1 |s1 , ..., sn ; AiV¯ × V

X √ + 3eiθ cn+3,n12 +1,n21 |s1 , ..., sn ; AiX × X¯ s′1 ,...,s′n

X √ cn+3,n12 ,n21 +2 |s1 , ..., sn ; AiX¯˜ × X˜ , − 3eiθ

(3.85)

s′1 ,...,s′n

and H12 |s1 , ..., sn ; AiV¯ × V =

X

s′1 ,...,s′n

−

cn+3,n12 ,n21 |s1 , ..., sn ; AiV¯ × V

X

s′1 ,...,s′n

cn+3,n12 +2,n21 +1 |s1 , ..., sn ; AiV × V¯

X √ − 3eiθ cn+3,n12 +2,n21 |s1 , ..., sn ; AiX × X¯ s′1 ,...,s′n

X √ cn+3,n12 ,n21 +1 |s1 , ..., sn ; AiX¯˜ × X˜ . + 3eiθ

(3.86)

s′1 ,...,s′n

4. Spectral analysis Spectral studies in planar N = 4 SYM have shown the systematic presence of degenerate pairs of states of opposite “parity”, where parity is the Z2 symmetry associated with complex conjugation of the SU (N ) gauge group [11, 13, 14, 31]. These degeneracies persist at higher loops, but are lifted by non-planar corrections. This phenomenon is naturally explained by the integrable structures of planar N = 4 SYM: the theory admits higher conserved charges that are parity-odd and map the degenerate eigenstates. In some models it is even possible to prove that the existence of parity pairs is a sufficient condition for integrability [31]. The upshot is that in N = 4 SYM the existence of parity pairs is one of the many pieces of evidence for the complete integrability of the theory. With this precedent in mind, we can look forward to a similar analysis in N = 2 SCQCD and in N = 1 SQCD. In this section we determine the low-lying spectrum of the one-loop dilation operator of both theories, in the closed non-compact subsectors that were used to uplift the full Hamiltonian. 4.1 N = 2 SCQCD We start our analysis with N = 2 SCQCD and with the more general quiver theory that interpolates between the Z2 orbifold of N = 4 SYM and N = 2 SCQCD. For background material we refer the reader to [34, 35, 32] where this model was studied extensively.

– 16 –

4.1.1 Parity The first thing we need to do is define a meaningful parity operation. We will take N = 4 SYM as our starting point where parity amounts to conjugation of the SU (N ) gauge group. Under parity, the Lie algebra generators transform as T ab → −(T ab )∗ = −T ba ,

(4.1)

where we have used hermiticity to trade conjugation by transposition. Now, as reviewed in [33,34] N = 2 SCQCD can be thought of as a limit of a two-parameter (g, gˇ) quiver theory with gauge group SU (Nc ) × SU (Ncˇ) (with Ncˇ ≡ Nc ): one has N = 2 SCQCD at gˇ = 0 and the Z2 orbifold of N = 4 SYM at g = gˇ. Starting from N = 4 SYM with gauge group SU (2Nc ) the Z2 orbifold theory is obtained by the projection     Aaαα˙ b 0 φa b 0 , Z= , Aαα˙ =  (4.2) 0 Aˇaαˇ α˙ ˇb 0 φˇaˇ ˇb     ˆ λaIb 0 0 ǫIˆJˆ ψ aaˇJ  , λˆ =  , λI =  (4.3) I ˇ aˇ ˜ˇbJˆ 0 λ ǫ 0 ψ ˆ ˆ ˇ b IJ Ib   0 QIaIˇ ˆa  , (4.4) XI Iˆ =  ˇbJˆJ ¯ −ǫIJ ǫ ˆ ˆ Q 0 IJ

b

where I, Iˆ = 1, 2. The parity operation described above implies the following transformations. For the fields in the vector multiplets, Aaαα˙ b ↔ −Abαα˙ a

ˇ Aˇaαˇ α˙ ˇb ↔ −Aˇbαα˙ aˇ

λaIb ↔ −λbIa

ˇ aˇ ˇ ↔ −λ ˇˇb λ Iˇ a Ib

φab ↔ −φba ,

ˇ φˇaˇˇb ↔ −φˇbaˇ ,

(4.5)

and analogous expressions for the conjugate fields. For the fields in the hypermultiplets, ˇ b ψIaˆˇb ↔ −ψ˜Ia ˆ

¯ˇb , QaI Iˆˇb ↔ Q ˆ I Ia

(4.6)

and analogous expressions for the conjugate fields. These transformations remain a symmetry also away from the orbifold point (that is, for arbitrary (g, gˇ)), as can be easily checked by inspection of the Lagrangian (see e.g. [34] for the explicit expression of the Lagrangian). This implies that the parity operation commutes with the dilation operator to all loops. Its action on single-trace states is then given by P |A1 . . . AL i = (−1)L+k(k+1)/2 |AL . . . A1 i , ¯˜ Q ↔ −Q. ˜ ψ¯ ↔ ψ, ¯ where k is the number of fermions and we replace ψ ↔ ψ,

– 17 –

(4.7)

4.1.2 Diagonalization We consider the SU (1, 1) × SU (1|1) × SU (1|1) × U (1) subsector defined in [32]. We focus on SU (1, 1) primaries (since descendants have the same anomalous dimensions) and take with no loss of generalities r ≥ 0. Table 2 corresponds to states with maximal r-charge. This ¯ and is therefore identical to the SU (1, 1) subsector subsector is made exclusively out of {D k λ} used in [31] to obtain the complete dilation operator of N = 4 SYM. Being a subsector of N = 4, it is integrable and, indeed, our results indicate that degenerate states with opposite parity show up consistently at each stage of the diagonalization. The notation Pn (x) denotes the roots of a polynomial of order n, we will not write the polynomial explicitly because we are really interested in the amount of parity pairs and not in the actual values of the energies. We will denote by Pn (x) all the polynomials of order n we encounter, even if they are different from each other. L

r

∆0

δ∆P [2gY2 M N/π 2 ]

3

3 2

7.5

5± 4 133 ± 96 761 ± 761 − , 560 480 179 ± 120 5± 4

9.5 10.5 11.5 4

2

8 9 10 11

5

5 2

9.5 10.5 11.5

√

1 (73 ± 37)− 48 19 ± 133 ± , 96 12 ± P3 (x)− , 761 480

√

1 (73 ± 37)+ 48 7 ± 19 ± , 12 4 √ − 1 P3 (x) , 24 (43 ± 5)±

Paired eigenvalues: ∼ 69 %

Table 2: SU (1, 1) primaries with maximal r-charge (r = L2 ) in the SU (1, 1)×SU (1|1)×SU (1|1)×U (1) sector of N = 2 SCQCD. We have omitted the one-dimensional subspaces where there is no room for a parity pair.

Being identical to the analogous N = 4 SYM sector, we cannot use the results of Table 2 as a test for integrability. The true dynamics of N = 2 SCQCD is encoded in subspaces ¯ Our results are where the r-charge is not maximal. For this we need states with Q and Q. presented in Table 3. As opposed to the results of Table 2 the presence of parity pairs here is less systematic. More insight is obtained if we also look at the Z2 orbifold (ˇ g = g). For the the orbifold theory (and for the whole interpolating theory with general gˇ, g) we have an SU (2)L symmetry not present in N = 2 SCQCD, so to make the analysis more transparent we restrict the diagonalization to SU (2)L singlets. Our results for the Z2 orbifold are shown in Table 4. As in the case with maximal r-charge, parity pairs show up consistently. This is again expected because this theory is known to be integrable [36].

– 18 –

L

r

∆0

3

1 2

4.5

δ∆P [2gY2 M N/π 2 ] 3− 3− 3+ , , 8 4 √ 4 √ 1 ± 31)− , 32 (21 ± 57)+ 25 − 25 − 25 + , , √24 −24 1 48 √ 1 (81 ± 561) (83 ± 409)+ , 96 96 3± 3− , 8 4 ± 15 + 15 + 1 , , 16 √ 16 √ − 1 1 (21 ± 57) (21 ± 57)− , 32 32 √ 1 (16 ± 31)+ , 85 , 0+ 24 √ 15 + 1 1− , 16 , 32 (21 ± 57)− 5 ± 9 − 25 + , 8 , 24 4 √ √ + 1 1 (16 ± 6) , 96 (81 ± 561)+ 16 √ 1 (83 ± 409)− 96 − 1± , 1+ , 15 16 √ 1 (21 ± 57)+ 32

15 − 1 , 24 (16 16

5.5 6.5 4

5

0

6

6

1

7

1 2

5

6.5

Paired eigenvalues: 16 %

Table 3: SU (1, 1) primaries with 0 ≤ r < N = 2 SCQCD.

L 2

in the SU (1, 1) × SU (1|1) × SU (1|1) × U (1) sector of

L

r

∆0

δ∆P [2gY2 M N/π 2 ]

3

1 2

4.5

1+ 3− 3− 3− , 4 , 4 , 4 2 3± 4 3 − 7 − 25 − 1 + , , 24 , 2 4 8 √ 15 ± 1 , (27 ± 57)± 16 32 3+ 25 − 25 − 25 − , 24 , 24 , 24 4 5 ± 15 ± 25 ± , , 4 16 √ 24 ± 1 (93 ± 249) 96 √ 1− 1− 1 , 2 , 8 (5 ± 13)− 2 3± 3± , 4 4 + 7 25 + 1 − 1 − , 24 , 2 , 2 8 √ 1 (5 ± 5)+ 8 3± 5± 5± 7± , 4 , 8 , 8 4 √ 15 ± 15 ± 1 , 16 , 4 (3 ± 2)± 16 √ √ 1 1 (27 ± 57)± , 32 (27 ± 57)± 32

5.5 6.5

4

0

5 6

Paired eigenvalues: ∼ 68 %

Table 4: SU (1, 1) primaries with 0 ≤ r < L2 in the SU (1, 1) × SU (1|1) × SU (1|1) × U (1) sector of the orbifold theory (ˇ g = g). We have restricted the diagonalization to SU (2)L singlets.

Finally we look at how some sample parity pairs of the orbifold theory evolve when we move away from the orbifold point. Our results are shown in Table 5. We see that for arbitrary values of κ ≡ gˇ/g the pairs are lifted and they are not in general recovered in the SCQCD

– 19 –

L

r

3

1 2

3 3 3 3

1 2

1 2

1 2

1 2

∆0 5.5 5.5 6.5 6.5 6.5

4

0

6

4

0

6

4

0

6

4

0

6

κ=1 √ + 57) √ + 57) √ − 57) √ − 57)

1 (27 32 1 (27 32 1 (27 32 1 (27 32

5 4 5 4

1 (93 96 1 (93 96 1 (93 96 1 (93 96

+ + − − 5 4 5 4 5 8 5 8

1 (3 4 1 (3 4 1 (3 4 1 (3 4

+ + − −

√ √

√ √

κ = 0.7

κ = 0.3

κ=0

0.97

0.94

15 16

0.95

0.90

0.39

0.08

0

0.47

0.30

1 4

1 (27 32

1 (81 96 1 (83 96 1 (81 96 1 (83 96

+

√

Yes 57)

Yes No

√

No 561)

Yes

409)

Yes

561)

Yes

409)

Yes

1.12

1.09

1.11

1.08

249)

0.81

0.63

249)

0.82

0.68

249)

0.48

0.09

0

No

249)

0.57

0.31

1 4

No

1.08

1.01

1

Yes

1.06

1.01

1

Yes

0.45

0.29

No

Yes

√ √

√ √

+

SCQCD

+ − −

√

√ √

0.49

0.28

2)

0.74

0.54

2)

0.81

0.66

1 4 1 4 1 2 5 8

2)

0.26

0.06

0

No

2)

0.30

0.08

0

No

No No

Table 5: Examples of evolution of Z2 orbifold pairs for different values of the parameter κ = ggˇ .

limit κ → 0. (Note that not all SU (2)L gauge singlets evolve to legitimate states of N = 2 SCQCD, which must obey the stronger condition of being SU (Nf ) singlets. In the last column of Table 5 we indicate whether the states belong or not to N = 2 SCQCD.) 4.2 N = 1 SQCD We now repeat the same analysis for N = 1 SQCD. Inspired by the transformation used in the N = 2 theory, we define the following parity operation. For the fields in the vector multiplet, Aaµb ↔ −Abµa

λab ↔ −λba ,

(4.8)

and analogous expressions for the conjugate fields. For the chiral multipets, ˜˜ıa , Qai ↔ −Q

ψ ai ↔ −ψ˜˜ıa ,

(4.9)

and analogous expressions for the conjugate fields. Again, these transformations are a symmetry of the Lagrangian and therefore commute with the dilation operator to all loops. The action on single-trace states is given by P |A1 . . . AL i = (−1)L+k(k+1)/2 |AL . . . A1 i ,

– 20 –

(4.10)

¯˜ Q ↔ Q, ˜ ψ¯ ↔ ψ, ˜ where k is the number of fermions and we make the replacements ψ ↔ ψ, ¯ ¯ ˜ Q ↔ Q. Our results for the diagonalization of generalized single-trace operators of length L ≤ 5 are shown in Table 6. We restrict to states with r-charge 0 < r < L. (We omit the sectors with r = L and r = 0, which are spanned respectively by {λk } and {F¯k }. These sectors are isomorphic to the analogous sectors in N = 4 SYM and thus inherit their integrability.) L

r

∆0

3

1

4.5 5.5 6.5

2

4 5

4

1

6.5 7.5

2

6 7

3

5.5 6.5

5

3

6.5

δ∆P [gY2 M N/π 2 ] 3− 1 (67 ± 96 25 − 1 , (81 24 96

4 √

,

3+ 8 −

457) , P3 (x)+ √ ± 473)− , P3 (x)+

25 ± 48 3 − 9 + , 16 16 √ 7 − 7 − 13 + 1 , 24 , 16 , 24 (13 ± 39)+ 16 3+ 9− , 8 4 P3 (x)− , P4 (x)+ √ √ 1 1 (9 ± 37)+ , 96 (67 ± 457)+ , P3 (x)− 16 9 ± , 0± 16 √ + 1 (81 ± 473) , P3 (x)− , P7 (x)+ , P8 (x)− 96 25 ± 48 7 − 13 + , 16 16 √ 1 11 + 17 − , , (71 ± 553)+ , P3 (x)− 16 16 96 1 (9 16

±

√

37)− ,

9 ± 16

Paired eigenvalues: ∼ 13 %

Table 6: SU (1, 1) primaries with 0 < r < L in the SU (1, 1) × U (1|1) sector of N = 1 SQCD.

The results are qualitatively similar to the ones for N = 2 SCQCD: there are a few parity pairs, but their presence is not as striking and systematic as in N = 4 SYM.

5. Discussion We have used superconformal invariance to constrain the planar one-loop dilation operator for N = 1 SQCD. The structure of the calculation is similar to the one in N = 2 SCQCD [32], and leads to the same surprising result: the one-loop Hamiltonian is uniquely fixed by symmetry. We have worked in the “electric” description of the theory, at the Banks-Zaks fixed point near the upper edge of the conformal window. It would be interesting to apply the same strategy to the dual magnetic theory, at the Banks-Zaks fixed point in the lower edge of the conformal window. The recent discovery [30] that in the scalar sector of N = 1 SQCD the planar one-loop dilation operator is captured by the integrable Ising chain in transverse magnetic field was one of the motivations of our work. A skeptic may point out that this identification was kinematically foreordained, since no other structure is allowed (though the precise value of the transverse magnetic field does require dynamical input). The same could be said about

– 21 –

the SU (2) scalar sector of N = 4 SYM, which is a priori guaranteed to take the form of the Heisenberg spin chain, but not (say) for the SO(6) sector considered in the seminal paper [10]. More convincing tests of one-loop integrability require the complete Hamiltonian, which we have obtained in this paper. Our analysis indicates that the presence of degenerate parity pairs, a hallmark of integrability, is not as systematic in N = 1 SQCD (or in N = 2 SCQCD) as it is in N = 4 SYM. These preliminary results are not particularly encouraging, but should of course be taken cum grano salis. If the complete theory turns out to be non-integrable, we can still ask whether there is scope for integrability in some closed subsectors. The question is really about all-loop integrability, since at one loop some simple sectors may be “accidentally” integrable. This may well be the case for the scalar sector of [30], which is only closed to lowest order: at higher orders the scalars mix with every other state, so either the whole theory turns out to be integrable or the integrability of the scalar spin chain is a one-loop accident. More promising is the situation in the sectors that happen to coincide at one-loop with analogous sectors of N = 4 SYM, thus inheriting its integrability properties, but that remain closed to all loops. k λ , Dk F The largest such sector is the SU (2, 1|1) sector spanned by the letters {D+ α˙ + +α˙ ++ }. While at one loop its Hamiltonian coincides with that of N = 4 SYM, it will start differing from it at sufficiently high order. It will be very interesting to investigate whether integrability is preserved.7 The SU (2, 1|1) sector exists of course also in the dual magnetic theory, so its integrability may allow to interpolate across the whole conformal window. We look forward to future investigations of this scenario.

Acknowledgements It is a pleasure to thank David Poland for useful conversations. The work of P.L. and L.R. was supported in part by NSF grant PHY-1066293. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

7

The same can be asked for the SU (2, 1|2) sector of the N = 2 interpolating theory (in particular of N = 2 k k k SCQCD) consisting of the letters {D+ α ˙ φ , D+α ˙ λI + , D+α ˙ F++ }. Here the question is even sharper since on the φ vacuum the two-body magnon S-matrix is completely fixed by symmetry (up to the overall dynamical phase) to be the SU (2|2) S-matrix of [20], which automatically satisfies the Yang-Baxter equation. Still, integrability is by no means obvious, since factorization of the n-body S-matrix is a stronger condition than Yang-Baxter. If this sector turns out to be integrable to all loops, its difference from the analogous sector of N = 4 SYM would be fully encoded in the expression of the dynamical phase and of the magnon dispersion relation. The difference with N = 4 SYM should start appearing at three loops [37].

– 22 –

A. N = 1 superconformal multiplets In this appendix we summarize some basic facts about N = 1 superconformal representation theory [9]. A generic long multiplet A∆ r(j1 ,j2 ) is generated by the action of 4 Poincaré super¯ α˙ on a superconformal primary which by definition is annihilated by all charges Qα and Q the conformal supercharges S. In Table 7 we have summarized the possible shortening and semi-shortening conditions.

Shortening Conditions B B¯ Bˆ C C¯ Cˆ

D ¯ D

Qα |rih.w. = 0 ¯ α˙ |rih.w. = 0 Q B ∩ B¯

Multiplet

j1 = 0

∆=

j2 = 0

∆=

j1 , j2 , r = 0

∆=0 ∆ = 2 − 23 r

h.w.

(Q) |ri = 0 for j1 = 0 α ˙ β˙ ¯ h.w. ǫ Q ˙ |ri =0 β

B ∩ C¯ B¯ ∩ C

1

∆ = 2 + 2j2 +

α˙

¯ 2 |rih.w. = 0 for j2 = 0 (Q) C ∩ C¯

Br(0,j2 ) B¯r(j ,0)

∆ = 2 + 2j1 − 32 r

ǫαβ Qβ |rih.w. =0 α 2

− 23 r 3 2r

3 2r

∆ = 2 + 23 r 3 2r

= (j1 − j2 )

j1 = j2 =

∆ = 2 + j1 + j2

0, − 23 r = j2 + 0, 32 r = j1 + 1

1

∆= ∆=

− 23 r = 3 2r = 1

1 + j2 + j1

Bˆ

Cr(j1 ,j2 ) Cr(0,j2 ) C¯r(j ,j ) 1

2

C¯r(j1 ,0) Cˆ(j ,j ) 1

2

D(0,j2 ) ¯ (j ,0) D 1

Table 7: Possible shortening conditions for the N = 1 superconformal algebra.

B. Oscillator Representation In this appendix we describe the oscillator representation of the N = 1 superconformal algebra SU (2, 2|1). We introduce two sets of bosonic oscillators (aα , a†α ), (bα˙ , b†α˙ ) and one fermionic oscillator (c, c† ), where (α, α) ˙ are Lorentz indices. In addition we will need three more “auxiliary” fermionic operators (di , d†i ), i = 1, 2, 3. The non-zero (anti)commutation relations are [aα , a†β ] = δβα , [b

α˙

, b†β˙ ] †

=

δβα˙˙ ,

{c, c } = 1 ,

{di , d†j }

= δij .

In this oscillator representation the generators of SU (2, 2|1) are given by

– 23 –

(B.1) (B.2) (B.3) (B.4)

1 Lβα = a†β aα − δβα a†γ aγ , 2 1 α ˙ † α ˙ L˙ β˙ = bβ˙ b − δβα˙˙ b†γ˙ bγ˙ , 2 1 1 r = c† c − d†1 d1 − d†2 d2 − d†3 d3 , 3 3 1 † γ 1 † γ˙ D = 1 + aγ a + bγ˙ b , 2 2 1 † γ 1 † γ˙ 1 † 1 1 3 C = 1 − aγ a + bγ˙ b − c c − d†1 d1 − d†2 d2 − d†3 d3 , 2 2 2 2 2 2 ¯ α˙ = b† c† , Q α˙ α˙ ¯ S = bα˙ c ,

Qα = a†α c , α

† α

S =c a ,

Pαβ˙ = a†α b†β˙ ,

K

αβ˙

α β˙

=a b ,

(B.5) (B.6) (B.7) (B.8) (B.9)

(B.10) (B.11) (B.12)

Here C is a central charge that must kill any physical state. It could be eliminated from the algebra by redefining r → r + 32 C, but it is useful for implementing the harmonic action so we keep it. B.1 Vector multiplets V and V¯ We define a vacuum state |0i annihilated by all the lowering operators. Then we identify D k F = (a† )k+2 (b† )k |0i , k

† k+1

D λ = (a )

† k †

(b ) c |0i ,

(B.13) (B.14)

and D k F¯ = (a† )k (b† )k+2 c† d†3 |0i , ¯ = (a† )k (b† )k+1 d† |0i . Dk λ 3

(B.15) (B.16)

B.2 Chiral multiplets X and X¯ Similarly, for the chiral multiplets we identify D k Q = (a† )k (b† )k c† d†1 |0i , k

† k+1

D ψ = (a )

(b† )k d†1 |0i ,

(B.17) (B.18)

and ¯ = (a† )k (b† )k d† d† |0i , Dk Q 1 2 D k ψ¯ = (a† )k (b† )k+1 c† d†1 d†2 |0i .

– 24 –

(B.19) (B.20)

B.3 Two-letter Superconformal Primaries

By demanding that they are annihilated by all the conformal supercharges and by the appropriate combinations of Poincaré supercharges, we have worked out the expressions for the superconformal primaries of the irreducible modules that appear on the right hand side of the tensor products (2.10–2.19). The grassmannOps.m oscillator package by Jeremy Michelson and Matthew Headrick was extremely useful for this task. We simply quote the results: V × V: B¯2(0,0) = λ+ λ− − λ− λ+ , B¯2(1,0) = λ+ λ+ ,    q−2 X (−1)k q + 1 q−2 ˆ C( q+1 , q−2 ) = D q−k−2 λ+ D k F++ 2 2 q(q + 1) k + 2 k k=0    q−2 q 1 X (−1)q−k q − 2 D k F++ D q−k−2 λ+ . + q k+2 k k+1

(B.21) (B.22)

(B.23)

k=0

¯ F¯ ). For V¯ × V¯ the expressions are identical with (λ, F) replaced by (λ, V × X: B¯5 ( 1 ,0) = λ+ Q ,

(B.24)

3 2

Cˆ( q+1 , q−1 ) = 2

2

   q−1 X q+1 k q−1 (−1) D q−k−1 F++ D k Q k k k=0    q−1 X (−1)k q − 1 q +(q + 1) D q−k−1 λ+ D k ψ+ . k+1 k k

(B.25)

k=0

˜ and interchange the order of the fields ˜ ψ) For the X˜ × V primary we replace (Q,ψ) by (Q, (taking into account fermionic minus signs). V¯ × X : Cˆ( q , q+1 ) = 2

2

   q X q+1 k q ¯ ˙ Dk Q (−1) D q−k λ + k k k=0    q−1 X (−1)k q − 1 q + 1 −q D q−k−1 F¯+˙ +˙ D k ψ+ . k+1 k k

(B.26)

(B.27)

k=0

˜ and interchange the order of the fields. ˜ ψ) For the X˜ × V¯ primary we replace (Q,ψ) by (Q,

– 25 –

X¯ × V: Cˆ( q+1 , q ) = 2

2

   q X q+1 k q ¯ k λ+ (−1) D q−k QD k k+1 k=0    q−1 X q+1 (−1)k q − 1 D q−k−1 ψ¯+˙ D k F++ . +q k+2 k k+1

(B.28)

k=0

˜ and interchange the order of the fields. ˜ primary we replace (Q,ψ) by (Q, ˜ ψ) For the V × X¯ ¯ X¯ × V: ¯+ , ¯λ B− 5 (0, 1 ) = Q 3

Cˆ( q−1 , q+1 ) = 2

(B.29)

2

2

   q−1 X (−1)k q − 1 q

¯˙ D k ψ¯+˙ D q−k−1 λ + k k k=0    q−1 X q+1 (−1)q−k q − 1 ¯ k F¯ ˙ ˙ . D q−k−1 QD + ++ q+1 k k+2 k+1

(B.30)

k=0

˜ and interchange the order of the fields. ˜ primary we replace (Q,ψ) by (Q, ˜ ψ) For the V¯ × X¯ X¯ × X : Cˆ( q , q ) = 2 2

   q X q q ¯ kQ (−1)k D q−k QD k k k=0    q−1 X q (−1)k q − 1 D q−k−1 ψ¯+˙ D k ψ+ . +q k+1 k k

(B.31)

k=0

This primary is gauge contracted. For the flavor contracted X˜¯ × X˜ primary we replace (Q,ψ) ˜ ˜ ψ). by (Q, ˜: X˜ × X¯ Cˆ( q , q ) 2 2

   q X q ¯˜ k q ˜ kQ = (−1) D q−k QD k k k=0    q−1 X q (−1)k q − 1 D q−k−1 ψ˜+ D k ψ¯˜+˙ . +q k+1 k k

(B.32)

k=0

˜ ˜ ψ) This primary is gauge contracted. For the flavor contracted X × X¯ primary we replace (Q, by (Q,ψ) .

– 26 –

X˜ × X : ˜ B¯ 4 (0,0) = QQ

(B.33)

3

Cˆ( q+1 , q ) 2

2

   q X q+1 k q ˜ k ψ+ = (−1) D q−k QD k k+1 k=0    q X q q+1 (−1)q−k − D k ψ˜+ D q−k Q . k k+1

(B.34)

k=0

¯˜ (also for the conjugates). ˜ primary we interchange (Q,ψ) ˜ ¯ ψ) For the X¯ × X¯ by (Q, ¯ V × V: Cˆ( q , q ) = 2 2

   q−1 X (−1)k q − 1 q

¯˙ D q−k−1 λ+ D k λ + k k k=0    q−2 X (−1)q−k q1 q − D k F++ D q−k−2 F¯+˙ +˙ . q k k+2 k+1

(B.35)

k=0

¯ F¯ ) . For the V¯ × V primary we replace (λ,F) by(λ,

References [1] N. Seiberg, Electric - magnetic duality in supersymmetric nonAbelian gauge theories, Nucl.Phys. B435 (1995) 129–146, [hep-th/9411149]. [2] I. R. Klebanov and J. M. Maldacena, Superconformal gauge theories and non-critical superstrings, Int.J.Mod.Phys. A19 (2004) 5003–5016, [hep-th/0409133]. [3] A. Fotopoulos, V. Niarchos, and N. Prezas, D-branes and SQCD in non-critical superstring theory, JHEP 10 (2005) 081, [hep-th/0504010]. [4] S. K. Ashok, S. Murthy, and J. Troost, D-branes in non-critical superstrings and minimal super Yang-Mills in various dimensions, Nucl. Phys. B749 (2006) 172–205, [hep-th/0504079]. [5] F. Bigazzi, R. Casero, A. L. Cotrone, E. Kiritsis, and A. Paredes, Non-critical holography and four-dimensional CFT’s with fundamentals, JHEP 10 (2005) 012, [hep-th/0505140]. [6] S. Murthy and J. Troost, D-branes in non-critical superstrings and duality in N = 1 gauge theories with flavor, JHEP 10 (2006) 019, [hep-th/0606203]. [7] C. Romelsberger, Counting chiral primaries in N = 1, d=4 superconformal field theories, Nucl.Phys. B747 (2006) 329–353, [hep-th/0510060]. [8] C. Romelsberger, Calculating the Superconformal Index and Seiberg Duality, arXiv:0707.3702. [9] F. Dolan and H. Osborn, Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories, Nucl.Phys. B818 (2009) 137–178, [arXiv:0801.4947].

– 27 –

[10] J. Minahan and K. Zarembo, The Bethe ansatz for N=4 superYang-Mills, JHEP 0303 (2003) 013, [hep-th/0212208]. [11] N. Beisert, C. Kristjansen, and M. Staudacher, The dilatation operator of N = 4 super Yang-Mills theory, Nucl. Phys. B664 (2003) 131–184, [hep-th/0303060]. [12] N. Beisert, The complete one loop dilatation operator of N=4 superYang-Mills theory, Nucl.Phys. B676 (2004) 3–42, [hep-th/0307015]. [13] N. Beisert, The su(2|3) dynamic spin chain, Nucl. Phys. B682 (2004) 487–520, [hep-th/0310252]. [14] B. I. Zwiebel, N=4 SYM to two loops: Compact expressions for the non-compact symmetry algebra of the su(1,1|2) sector, JHEP 0602 (2006) 055, [hep-th/0511109]. [15] C. Sieg, Superspace computation of the three-loop dilatation operator of N=4 SYM theory, Phys.Rev. D84 (2011) 045014, [arXiv:1008.3351]. [16] B. I. Zwiebel, From Scattering Amplitudes to the Dilatation Generator in N=4 SYM, arXiv:1111.0083. * Temporary entry *. [17] N. Beisert and M. Staudacher, The N=4 SYM integrable super spin chain, Nucl.Phys. B670 (2003) 439–463, [hep-th/0307042]. [18] M. Staudacher, The Factorized S-matrix of CFT/AdS, JHEP 0505 (2005) 054, [hep-th/0412188]. [19] N. Beisert and M. Staudacher, Long-range psu(2,2|4) Bethe Ansatze for gauge theory and strings, Nucl.Phys. B727 (2005) 1–62, [hep-th/0504190]. In honor of Hans Bethe. [20] N. Beisert, The su(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945, [hep-th/0511082]. [21] N. Beisert, B. Eden, and M. Staudacher, Transcendentality and Crossing, J.Stat.Mech. 0701 (2007) P01021, [hep-th/0610251]. [22] N. Gromov, V. Kazakov, and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar N=4 Supersymmetric Yang-Mills Theory, Phys.Rev.Lett. 103 (2009) 131601, [arXiv:0901.3753]. [23] N. Beisert et. al., Review of AdS/CFT Integrability: An Overview, arXiv:1012.3982. [24] A. Belitsky, G. Korchemsky, and D. Mueller, Integrability in Yang-Mills theory on the light cone beyond leading order, Phys.Rev.Lett. 94 (2005) 151603, [hep-th/0412054]. [25] A. Belitsky, G. Korchemsky, and D. Mueller, Integrability of two-loop dilatation operator in gauge theories, Nucl.Phys. B735 (2006) 17–83, [hep-th/0509121]. [26] A. Belitsky, G. Korchemsky, and D. Mueller, Towards Baxter equation in supersymmetric Yang-Mills theories, Nucl.Phys. B768 (2007) 116–134, [hep-th/0605291]. [27] G. Korchemsky, Review of AdS/CFT Integrability, Chapter IV.4: Integrability in QCD and N < 4 SYM, arXiv:1012.4000. * Temporary entry *. [28] G. Veneziano, Some Aspects of a Unified Approach to Gauge, Dual and Gribov Theories, Nucl.Phys. B117 (1976) 519–545.

– 28 –

[29] T. Banks and A. Zaks, On the Phase Structure of Vector-Like Gauge Theories with Massless Fermions, Nucl.Phys. B196 (1982) 189. [30] D. Poland and D. Simmons-Duffin, N=1 SQCD and the Transverse Field Ising Model, arXiv:1104.1425. [31] N. Beisert, The Dilatation operator of N=4 super Yang-Mills theory and integrability, Phys.Rept. 405 (2005) 1–202, [hep-th/0407277]. Ph.D. Thesis. [32] P. Liendo, E. Pomoni, and L. Rastelli, The Complete One-Loop Dilation Operator of N=2 SuperConformal QCD, arXiv:1105.3972. [33] A. Gadde, E. Pomoni, and L. Rastelli, The Veneziano Limit of N = 2 Superconformal QCD: Towards the String Dual of N = 2 SU(N(c)) SYM with N(f ) = 2 N(c), arXiv:0912.4918. [34] A. Gadde, E. Pomoni, and L. Rastelli, Spin Chains in N=2 Superconformal Theories: From the Z2 Quiver to Superconformal QCD, arXiv:1006.0015. [35] A. Gadde and L. Rastelli, Twisted Magnons, arXiv:1012.2097. [36] N. Beisert and R. Roiban, The Bethe ansatz for Z(S) orbifolds of N = 4 super Yang- Mills theory, JHEP 11 (2005) 037, [hep-th/0510209]. [37] E. Pomoni and C. Sieg, From N=4 gauge theory to N=2 conformal QCD: three-loop mixing of scalar composite operators, arXiv:1105.3487.

– 29 –